The first music shop I encountered was George Potter in Aldershot in 1963. They were mainly involved in providing instruments for military bands but hada few guitars and basses. It was there that I bought my first real bass, a Framus Star bass. The following year a brand new shop opened just 10 miles away in Guildford called Andertons, and the rest is history.
I also remember visiting Monkey Business in Romford when I happened to passing on business.
Not strictly a music shop but I bought my first real amp, a Linear 30, from Radio Component Specialists (RCS) in Croydon. That was followed by 4 Bakers Group 25 speakers which were fitted into a cabinet the same size as a Marshall 4 x12.
When I briefly worked in London in 1965 I used to break my journey from Waterloo to Islington at Leicester Square and walk up Tottenham Court Road gazing into Macaris and all the other music shops, especially the shop under Centre Point. Alas I had no spare money for any more gear then.